Dispersive estimates for inhomogeneous fourth-order Schrödinger operator in 3D with zero energy obstructions

Abstract

We study the $L^1-L^{\infty }$ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator $H={\Delta }^2-\Delta +V(x)$ with zero energy obstructions in ${\mathbf{R}}^3$. For the related propagator $e^{-itH}$, we prove that for $0<t\le 1$, then $e^{-itH}{P}_{ac}(H)$ satisfies the $|t{|}^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}{P}_{ac}(H)$ satisfies the $|t{|}^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_t$ such that $e^{-itH}{P}_{ac}(H)-F_t$ satisfies the $|t{|}^{-3/2}$- dispersive estimate.\,\, 3) if zero is a resonance and~/~or an eigenvalue of $H$, then there exists a time dependent operator $G_t$ such that $e^{-itH}{P}_{ac}(H)-G_t$ satisfies the $|t{|}^{-3/2}$- dispersive estimate. Here $F_t$ and $G_t$ satisfy $|t{|}^{-1/2}$-dispersive estimates.

32 pages